10,387 research outputs found
Lower large deviations for supercritical branching processes in random environment
Branching Processes in Random Environment (BPREs) are the
generalization of Galton-Watson processes where in each generation the
reproduction law is picked randomly in an i.i.d. manner. In the supercritical
regime, the process survives with a positive probability and grows
exponentially on the non-extinction event. We focus on rare events when the
process takes positive values but lower than expected. More precisely, we are
interested in the lower large deviations of , which means the asymptotic
behavior of the probability as
. We provide an expression of the rate of decrease of this
probability, under some moment assumptions, which yields the rate function.
This result generalizes the lower large deviation theorem of Bansaye and
Berestycki (2009) by considering processes where \P(Z\_1=0 \vert
Z\_0=1)\textgreater{}0 and also much weaker moment assumptions.Comment: A mistake in the previous version has been corrected in the
expression of the speed of decrease in the case without
extinctio
Upper large deviations for Branching Processes in Random Environment with heavy tails
Branching Processes in a Random Environment (BPREs) are a
generalization of Galton Watson processes where in each generation the
reproduction law is picked randomly in an i.i.d. manner. We determine here the
upper large deviation of the process when the reproduction law may have heavy
tails. The behavior of BPREs is related to the associated random walk of the
environment, whose increments are distributed like the logarithmic mean of the
offspring distributions. We obtain an expression of the upper rate function of
, that is the limit of
when . It depends on the rate function of the associated random
walk of the environment, the logarithmic cost of survival
and the polynomial decay
of the tail distribution of . We give interpretations of this rate
function in terms of the least costly ways for the process of
attaining extraordinarily large values and describe the phase transitions. We
derive then the rate function when the reproduction law does not have heavy
tails, which generalizes the results of B\"oinghoff and Kersting (2009) and
Bansaye and Berestycki (2008) for upper large deviations. Finally, we specify
the upper large deviations for the Galton Watson processes with heavy tails
Valuing entrepreneurial investments: the venture capitalists' approach.
Valuing high-growth, high-uncertainty firms, characterised by a unique business concept, significant growth opportunities, and/or no real positive cash flows to show the profit potential of the venture, is a major challenge faced by most venture capitalists (Gompers (1995)). Unlike for an investment in publicly traded securities for which there exists a well-defined pricing mechanism, it is difficult to find an objective valuation for the investment holdings of a venture capital fund. The valuation of individual unquoted investments is, thus, a very complicated process. subject to the discretion and judgment from the part of the venture capitalist. Recently, growing criticism and increasing interest are observed regarding the valuation of the private equity and venture capital portfolios of high-tech, high risk, high growth venture investments (EVCA (2001), Millner (2002), Blaydon & Horvath (2002)). Consequently, the underlying goal of the empirical analyses included in this paper corresponds exactly with revealing the valuation methodology operated by venture capitalists when determining or reconsidering the valuation for each venture investment held in portfolio.Cash flow; Firms; Investment; Investment portfolio; Investments; Opportunities; Portfolio; Pricing; Processes; Risk; Studies; Valuation; Valuation method; Venture capital;
Phenotypic diversity and population growth in fluctuating environment: a MBPRE approach
Organisms adapt to fluctuating environments by regulating their dynamics, and
by adjusting their phenotypes to environmental changes. We model population
growth using multitype branching processes in random environments, where the
offspring distribution of some organism having trait t\in\cT in environment
e\in\cE is given by some (fixed) distribution on \bbN.
Then, the phenotypes are attributed using a distribution (strategy)
on the trait space \cT. We look for the optimal strategy ,
t\in\cT, e\in\cE maximizing the net growth rate or Lyapounov exponent, and
characterize the set of optimal strategies. This is considered for various
models of interest in biology: hereditary versus non-hereditary strategies and
strategies involving or not involving a sensing mechanism. Our main results are
obtained in the setting of non-hereditary strategies: thanks to a reduction to
simple branching processes in random environment, we derive an exact expression
for the net growth rate and a characterisation of optimal strategies. We also
focus on typical genealogies, that is, we consider the problem of finding the
typical lineage of a randomly chosen organism.Comment: 21 page
Confinement by biased velocity jumps: aggregation of Escherichia coli
We investigate a linear kinetic equation derived from a velocity jump process
modelling bacterial chemotaxis in the presence of an external chemical signal
centered at the origin. We prove the existence of a positive equilibrium
distribution with an exponential decay at infinity. We deduce a hypocoercivity
result, namely: the solution of the Cauchy problem converges exponentially fast
towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot,
and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,
Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to
the null spaces of the collision operator and of the transport operator. From a
modelling viewpoint it is related to the observation that exponential
confinement is generated by a spatially inhomogeneous bias in the velocity jump
process.Comment: 15 page
Bayesian sequential change diagnosis
Sequential change diagnosis is the joint problem of detection and
identification of a sudden and unobservable change in the distribution of a
random sequence. In this problem, the common probability law of a sequence of
i.i.d. random variables suddenly changes at some disorder time to one of
finitely many alternatives. This disorder time marks the start of a new regime,
whose fingerprint is the new law of observations. Both the disorder time and
the identity of the new regime are unknown and unobservable. The objective is
to detect the regime-change as soon as possible, and, at the same time, to
determine its identity as accurately as possible. Prompt and correct diagnosis
is crucial for quick execution of the most appropriate measures in response to
the new regime, as in fault detection and isolation in industrial processes,
and target detection and identification in national defense. The problem is
formulated in a Bayesian framework. An optimal sequential decision strategy is
found, and an accurate numerical scheme is described for its implementation.
Geometrical properties of the optimal strategy are illustrated via numerical
examples. The traditional problems of Bayesian change-detection and Bayesian
sequential multi-hypothesis testing are solved as special cases. In addition, a
solution is obtained for the problem of detection and identification of
component failure(s) in a system with suspended animation
Mechanics of large folds in thin interfacial films
A thin film at a liquid interface responds to uniaxial confinement by
wrinkling and then by folding; its shape and energy have been computed exactly
before self contact. Here, we address the mechanics of large folds, i.e. folds
that absorb a length much larger than the wrinkle wavelength. With scaling
arguments and numerical simulations, we show that the antisymmetric fold is
energetically favorable and can absorb any excess length at zero pressure.
Then, motivated by puzzles arising in the comparison of this simple model to
experiments on lipid monolayers and capillary rafts, we discuss how to
incorporate film weight, self-adhesion and energy dissipation.Comment: 5 pages, 3 figure
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